§ 2.5

§ 2.5

The Point-Slope Form of the Equation of a Line

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 2.5

Point-Slope Form 144

Point-Slope Form of the Equation of a Line

The point-slope equation of a nonvertical line with slope m that passes through the point is 11, yx

11 xxmyy



EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Write the point-slope form and then the slope-intercept form of the equation of the line with slope -3 that passes through the point (2,-4).

Substitute the given values 11 xxmyy

234 xy

634 xy23 xy

Distribute

Subtract 4 from both sides

234 xy Simplify

This is the equation of the line in point-slope form.

This is the equation of the line in slope-intercept form.



EXAMPLEEXAMPLE

SOLUTIONSOLUTION

First I must find the slope of the line. That is done as follows:

2

5

10

23

46

m

Write the point-slope form and then the slope-intercept form of the equation of the line that passes through the points (2,-4) and (-3,6).



Now I can find the two forms of the equation of the line. In find the point-slope form of the line, I can use either point provided. I’ll use (2,-4).

Substitute the given values 11 xxmyy

224 xy

424 xyxy 2

Distribute


224 xy Simplify



CONTINUECONTINUEDD


Equations of Lines 146

11 xxmyy

Equations of LinesStandard Form Ax + By = C

Slope-Intercept Form y = mx + b

Horizontal Line y = b

Vertical Line x = a

Point-slope Form


Deciding which form to use: 146

y = mx + b y – y1 = m(x – x1)Begin with the slope-intercept form if you know:

Begin with the point-slope form if you know:

The slope of the line and the y-intercept

or

Two points on the line, one of which is the y –intercept

The slope of the line and a point on the line other than the y-intercept

or

Two points on the line, neither of which is the y-intercept


Modeling Life Expectancy 147

EXAMPLE 3EXAMPLE 3

Go over example 3 See Figures 2.35(a) and 2.35(b)

Use the data points to write the slope-intercept form of the equation of this line. Use the linear function to predict the life expectancy of an American man born in 2020.

Find the slope of 0.215. The slope indicates that for each subsequent birth year, a man’s life expectancy is increasing by 0.215 years.

Use the point-slope form to write the equation (model).y – 70.0 = 0.215(x – 20)

Change to slope-intercept formy = 0.215x + 65.7

Life expectancy of men born in 2020f(60) = 0.215(60) + 65.7 = 78.6 years



Check PointsCheck Points

Do Check Point 3 on page 148 See Figure 2.36

Use the data points to write the slope-intercept form of the equation of this line. (round to 2 decimal places). Use the linear function to predict the life expectancy of an American woman born in 2020.



Check Points, Check Points, continuedcontinuedDo Check Point 3 on page 148 See Figure 2.36

Use the data points to write the slope-intercept form of the equation of this line. (round to 2 decimal places). Use the linear function to predict the life expectancy of an American woman born in 2020.

Public Tuition: In 2005, the average cost of tuition and fees at public four-year colleges was $6130, and in 2010 it was $7610. Note that the known value for 2008 is $6530.

Solution: The line passes through (2005, 6.1) and (2010, 7.6). Find the slope.

20052010

61307610

in x change

yin change

m 2965

1480

Thus, the slope of the line is 296; tuition and fees on average increased by $296/yr.

Figure not in book

0510

61307610

in x change

yin change

m 2965

1480

Substitute 5 for 2005, 10 for 2010, and 8 for 2008.

Modeling Public Tuition

Modeling public tuition: Write the slope-intercept form of the of the line shown in the graph. What is the y-intercept and does it have meaning in this situation.

11 xxmyy

20052966130 xy

5934802966130 xy

587350296 xy



2965

1480

Modeling public tuition: Substitute 5 for 2005, 10 for 2010, and 8 for 2008.

11 xxmyy

52966130 xy

14802966130 xy

4650296 xy




Using the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the average cost of tuition and fees at public four-year colleges in 2008.

Substitute 2008 or 8 for x and compute y.

587350)2008(296 y 7018

587350296 xy

4650)8(296 y 7018

The model predicts that the tuition in 2008 will be $7018

Use the equation to predict the average cost of tuition and fees at public four-year colleges in 2015.

Substitute 2015 or15 for x and compute y.

587350)2015(296 y 9090

4650)15(296 y 9090


The model predicts that the tuition in 2015 will be $9090.

Write the slope-intercept form of the equation of the line shown in the graph. Use the equation to predict the median age of the U.S. population in 2020.

Solution: The line passes through (10, 30.0) and (30, 35.3). Find the slope.

(10, 30.0)

(30, 35.3)

1030

0.303.35

in x change

yin change

m 265.020

3.5

The slope indicates that each year the median age of the U.S. population is increasing by 0.265 year.

Modeling the Graying of America


265.020

3.5m


11 xxmyy 10265.00.30 xy

65.2265.00.30 xy

35.27265.0 xy

A linear equation that models the median age of the U.S. population, y, x years after 1970.


This is the equation of the line in slope-intercept form. (10, 30.0)

(30, 35.3)



265.020

3.5m


35.27265.0 xy

A linear equation that models the median age of the U.S. population, y, x years after 1970.

Use the equation to predict the median age in 2020. Because 2020 is 50 years after 1970, substitute 50 for x and compute y.

35.27)50(265.0 y 6.40The model predicts that the median age of the U.S. population in 2020 will be 40.6.

(10, 30.0)

(30, 35.3)



Parallel and Perpendicular Lines

Slope and Parallel Lines1) If two nonvertical lines are parallel, then they have the same slope.

2) If two distinct nonvertical lines have the same slope, then they are parallel.

3) Two distinct vertical lines, both with undefined slopes, are parallel.

y = 2x + 6 and y = 2x – 4 are parallel.

y = -4x +5 and y = -4x + 3 are parallel.



Slope and Perpendicular Lines1) If two nonvertical lines are perpendicular, then the product of their slopes is -1.

2) If the product of the slopes of two lines is -1, then the lines are perpendicular.

3) A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.

y = 2x + 6 and y = -(1/2)x – 4 are perpendicular.

y = -4x +5 and y = (1/4)x + 3 are perpendicular.


Parallel and Perpendicular Lines 148

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Write an equation of the line passing through (2,-4) and parallel to the line whose equation is y = -3x + 5.

Since the line I want to represent is parallel to the given line, they have the same slope. Therefore the slope of the new line is also m = -3. Therefore, the equation of the new line is:

y – (-4) = -3(x –2)

y + 4 = -3(x –2)

y + 4 = -3x + 6

y = -3x + 2

Substitute the given values

Simplify

Distribute



Parallel and Perpendicular Lines 149-150

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Write an equation of the line passing through (2,-4) and perpendicular to the line whose equation is y = -3x + 5.

The slope of the given equation is m = -3. Therefore, the slope of the new line is , since . Therefore, the using the slope m = and the point (2,-4), the equation of the line is as follows:

31 13 3

1

31



11 xxmyy

23

14 xy

23

14 xy

3

2

3

14 xy

43

2

3

1 xy

3

3

1

4

3

2

3

1 xy

3

12

3

2

3

1 xy

3

14

3

1 xy

CONTINUECONTINUEDD

Substitute the given values

Simplify

Simplify

Distribute


Common Denominators

Common Denominators

m = and the point (2,-4), 31


Parallel and Perpendicular Lines 149 -150

Do Check Point 4 on page 149Find the equation that passes through (-2, 5) and is parallel to the line y = 3x+1


Parallel and Perpendicular Lines 149 -150

Do Check Point 5 on page 150Find the slope and equation of a line that passes through (-2, -6) and is perpendicular to x+3y=12 Standard format must be changed



One line is perpendicular to another line if its slope is the negative reciprocal of the slope of the other line.

The following lines are perpendicular:

y = 2x + 6 and y = -(1/2)x – 4 are perpendicular.

y = -4x +5 and y = (1/4)x + 3 are perpendicular.



Two lines are parallel if they have the same slope.

The following lines are parallel:

y = 2x + 6 and y = 2x – 4 are parallel.

y = -4x +5 and y = -4x + 3 are parallel.

Example Modeling female officers

In 1995, there were 690 female officers in the Marine Corps, and by 2010 this number had increased to about 1110. Refer to graph in Figure 3.48 on page 214.

a)The slope of the line passing through (1995, 690) and (2010,1110) is

b)The number of female officers increased, on average by about 28 officers per year.

c)Estimate how many female officers there were in 2006.

19952010

6901110

m 28

Write the slope-intercept form of the of the line shown in the graph.

(1995, 690)

(2010, 1110)

//

11 xxmyy 199528690 xy

5586028690 xy5517028 xy

5517028 xyOR

9985517056168

55170)2006(28

y

y

1995200628690 y 9986901128 y

§ 2.5

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